Worked examples, in Risk & Portfolio Construction
Returns Distribution Analyzer: Examples
Short samples produce noisy higher moments; these scenarios are designed to show the direction of the change rather than any precise decimal. Skew measures asymmetry, excess kurtosis measures tail fatness relative to normal, and the Jarque-Bera statistic combines both into a normality test: a high statistic with a low p-value means non-normal returns. Compare what happens to these statistics as the series shifts from symmetric to skewed, and from thin-tailed to fat-tailed.
3 examples. Published May 26, 2026. By AI Fin Hub Research.
Worked examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
Symmetric series looks normal
A balanced series of paired gains and losses around a zero mean. No asymmetry and no fat tails.
Mean 0, skew 0, excess kurtosis minus 1.65, Jarque-Bera 1.36 (p = 0.51).
Returns
+/- 0.01, +/- 0.015, +/- 0.005, +/- 0.012, +/- 0.008, +/- 0.003
Bins
7
Skew is exactly zero because the series is symmetric, and the Jarque-Bera p-value of 0.51 fails to reject normality. The negative excess kurtosis reflects a flat, platykurtic shape from the small evenly-spaced sample, not a tail problem. This is the clean baseline.
- 2
Same calm, one crash day
Nine small moves around zero plus a single minus-8-percent shock. The kind of tail event that defines real return series.
Skew minus 2.19, excess kurtosis 3.32, Jarque-Bera 12.60 (p = 0.002).
Returns
0.005, 0.004, -0.003, 0.006, -0.002, 0.005, 0.003, -0.004, 0.005, -0.08
Bins
7
One crash day drives skew to minus 2.19 and excess kurtosis to 3.32, and Jarque-Bera now rejects normality at p = 0.002. The median is still positive, so a naive look at central tendency would miss the risk entirely. Higher moments, not the average, expose tail danger.
- 3
Mildly negative-skew series
A series with frequent small gains and a few larger losses, the typical fingerprint of a short-volatility or carry strategy.
Mean 0.002, skew minus 0.31, excess kurtosis minus 1.57, Jarque-Bera 1.19 (p = 0.55).
Returns
0.01, -0.02, 0.015, -0.005, 0.02, -0.01, 0.008, 0.012, -0.015, 0.005
Bins
5
Skew is mildly negative at minus 0.31, hinting at asymmetric losses, but with only ten observations Jarque-Bera cannot reject normality (p = 0.55). The lesson is statistical power: a real negative-skew profile may need dozens of observations before the normality test can flag it.
Patterns
Run the numbers next
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Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
- Playgrounds VaR Backtest — Kupiec & Christoffersen
Paste P&L + VaR series and run Kupiec POF, Christoffersen independence, and joint conditional-coverage tests. Likelihood-ratio χ² p-values.
Sources
- A Test for Normality of Observations and Regression Residuals , Jarque, C. M. and Bera, A. K., International Statistical Review (1987)
- The Variation of Certain Speculative Prices , Mandelbrot, B., Journal of Business (1963)
Keep the topic connected
- Learn Volatility
Volatility as the standard deviation of returns: realized vs implied, the annualization gotcha, and why volatility-of-volatility matters.
- Learn Expected Shortfall (CVaR)
Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.
- Learn Value at Risk (VaR)
Value at Risk: the loss threshold you'll exceed with probability α. Why historical VaR is brittle and what it doesn't tell you about the tail.